LMFDB - Embedded newform 1344.4.c.f.673.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.4
Root			$-1.04508 - 3.90029i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.9

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -1.12662i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -1.12662i q^{5} -7.00000 q^{7} -9.00000 q^{9} +16.9437i q^{11} -11.6634i q^{13} -3.37986 q^{15} -69.2532 q^{17} -87.3889i q^{19} +21.0000i q^{21} +49.5619 q^{23} +123.731 q^{25} +27.0000i q^{27} -178.161i q^{29} -147.229 q^{31} +50.8310 q^{33} +7.88634i q^{35} +241.116i q^{37} -34.9901 q^{39} -61.3207 q^{41} +283.169i q^{43} +10.1396i q^{45} -300.001 q^{47} +49.0000 q^{49} +207.760i q^{51} -44.4371i q^{53} +19.0891 q^{55} -262.167 q^{57} +260.018i q^{59} -151.277i q^{61} +63.0000 q^{63} -13.1402 q^{65} +635.185i q^{67} -148.686i q^{69} -51.2855 q^{71} -346.976 q^{73} -371.192i q^{75} -118.606i q^{77} +173.864 q^{79} +81.0000 q^{81} +524.089i q^{83} +78.0221i q^{85} -534.483 q^{87} +599.736 q^{89} +81.6437i q^{91} +441.686i q^{93} -98.4542 q^{95} +468.989 q^{97} -152.493i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 1.12662i	− 0.100768i	−0.998730	−	0.0503840i	$-0.983955\pi$
$5$	− 1.12662i	− 0.100768i	0.998730	−	0.0503840i	$-0.0160445\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	16.9437i	0.464428i	0.972665	+	0.232214i	$0.0745970\pi$
$11$	16.9437i	0.464428i	−0.972665	+	0.232214i	$0.925403\pi$
$12$	0	0
$13$	− 11.6634i	− 0.248834i	−0.992230	−	0.124417i	$-0.960294\pi$
$13$	− 11.6634i	− 0.248834i	0.992230	−	0.124417i	$-0.0397060\pi$
$14$	0	0
$15$	−3.37986	−0.0581784
$16$	0	0
$17$	−69.2532	−0.988022	−0.494011	−	0.869456i	$-0.664470\pi$
$17$	−69.2532	−0.988022	−0.494011	+	0.869456i	$0.664470\pi$
$18$	0	0
$19$	− 87.3889i	− 1.05518i	−0.849500	−	0.527589i	$-0.823096\pi$
$19$	− 87.3889i	− 1.05518i	0.849500	−	0.527589i	$-0.176904\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	49.5619	0.449320	0.224660	−	0.974437i	$-0.427873\pi$
$23$	49.5619	0.449320	0.224660	+	0.974437i	$0.427873\pi$
$24$	0	0
$25$	123.731	0.989846
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 178.161i	− 1.14082i	−0.821361	−	0.570408i	$-0.806785\pi$
$29$	− 178.161i	− 1.14082i	0.821361	−	0.570408i	$-0.193215\pi$
$30$	0	0
$31$	−147.229	−0.853002	−0.426501	−	0.904487i	$-0.640254\pi$
$31$	−147.229	−0.853002	−0.426501	+	0.904487i	$0.640254\pi$
$32$	0	0
$33$	50.8310	0.268138
$34$	0	0
$35$	7.88634i	0.0380867i
$36$	0	0
$37$	241.116i	1.07133i	0.844431	+	0.535664i	$0.179939\pi$
$37$	241.116i	1.07133i	−0.844431	+	0.535664i	$0.820061\pi$
$38$	0	0
$39$	−34.9901	−0.143664
$40$	0	0
$41$	−61.3207	−0.233578	−0.116789	−	0.993157i	$-0.537260\pi$
$41$	−61.3207	−0.233578	−0.116789	+	0.993157i	$0.537260\pi$
$42$	0	0
$43$	283.169i	1.00425i	0.864795	+	0.502126i	$0.167449\pi$
$43$	283.169i	1.00425i	−0.864795	+	0.502126i	$0.832551\pi$
$44$	0	0
$45$	10.1396i	0.0335893i
$46$	0	0
$47$	−300.001	−0.931056	−0.465528	−	0.885033i	$-0.654135\pi$
$47$	−300.001	−0.931056	−0.465528	+	0.885033i	$0.654135\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	207.760i	0.570435i
$52$	0	0
$53$	− 44.4371i	− 0.115168i	−0.998341	−	0.0575841i	$-0.981660\pi$
$53$	− 44.4371i	− 0.115168i	0.998341	−	0.0575841i	$-0.0183397\pi$
$54$	0	0
$55$	19.0891	0.0467995
$56$	0	0
$57$	−262.167	−0.609208
$58$	0	0
$59$	260.018i	0.573755i	0.957967	+	0.286877i	$0.0926172\pi$
$59$	260.018i	0.573755i	−0.957967	+	0.286877i	$0.907383\pi$
$60$	0	0
$61$	− 151.277i	− 0.317525i	−0.987317	−	0.158762i	$-0.949250\pi$
$61$	− 151.277i	− 0.317525i	0.987317	−	0.158762i	$-0.0507504\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−13.1402	−0.0250745
$66$	0	0
$67$	635.185i	1.15821i	0.815252	+	0.579106i	$0.196598\pi$
$67$	635.185i	1.15821i	−0.815252	+	0.579106i	$0.803402\pi$
$68$	0	0
$69$	− 148.686i	− 0.259415i
$70$	0	0
$71$	−51.2855	−0.0857249	−0.0428624	−	0.999081i	$-0.513648\pi$
$71$	−51.2855	−0.0857249	−0.0428624	+	0.999081i	$0.513648\pi$
$72$	0	0
$73$	−346.976	−0.556308	−0.278154	−	0.960536i	$-0.589723\pi$
$73$	−346.976	−0.556308	−0.278154	+	0.960536i	$0.589723\pi$
$74$	0	0
$75$	− 371.192i	− 0.571488i
$76$	0	0
$77$	− 118.606i	− 0.175537i
$78$	0	0
$79$	173.864	0.247611	0.123805	−	0.992307i	$-0.460490\pi$
$79$	173.864	0.247611	0.123805	+	0.992307i	$0.460490\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	524.089i	0.693088i	0.938034	+	0.346544i	$0.112645\pi$
$83$	524.089i	0.693088i	−0.938034	+	0.346544i	$0.887355\pi$
$84$	0	0
$85$	78.0221i	0.0995610i
$86$	0	0
$87$	−534.483	−0.658651
$88$	0	0
$89$	599.736	0.714290	0.357145	−	0.934049i	$-0.383750\pi$
$89$	599.736	0.714290	0.357145	+	0.934049i	$0.383750\pi$
$90$	0	0
$91$	81.6437i	0.0940504i
$92$	0	0
$93$	441.686i	0.492481i
$94$	0	0
$95$	−98.4542	−0.106328
$96$	0	0
$97$	468.989	0.490913	0.245457	−	0.969408i	$-0.421062\pi$
$97$	468.989	0.490913	0.245457	+	0.969408i	$0.421062\pi$
$98$	0	0
$99$	− 152.493i	− 0.154809i
$100$	0	0
$101$	1616.10i	1.59215i	0.605195	+	0.796077i	$0.293095\pi$
$101$	1616.10i	1.59215i	−0.605195	+	0.796077i	$0.706905\pi$
$102$	0	0
$103$	−1658.20	−1.58628	−0.793142	−	0.609037i	$-0.791556\pi$
$103$	−1658.20	−1.58628	−0.793142	+	0.609037i	$0.791556\pi$
$104$	0	0
$105$	23.6590	0.0219894
$106$	0	0
$107$	1076.84i	0.972915i	0.873704	+	0.486457i	$0.161711\pi$
$107$	1076.84i	0.972915i	−0.873704	+	0.486457i	$0.838289\pi$
$108$	0	0
$109$	142.750i	0.125440i	0.998031	+	0.0627199i	$0.0199775\pi$
$109$	142.750i	0.125440i	−0.998031	+	0.0627199i	$0.980023\pi$
$110$	0	0
$111$	723.347	0.618532
$112$	0	0
$113$	1398.19	1.16399	0.581993	−	0.813194i	$-0.302273\pi$
$113$	1398.19	1.16399	0.581993	+	0.813194i	$0.302273\pi$
$114$	0	0
$115$	− 55.8374i	− 0.0452771i
$116$	0	0
$117$	104.970i	0.0829446i
$118$	0	0
$119$	484.773	0.373437
$120$	0	0
$121$	1043.91	0.784306
$122$	0	0
$123$	183.962i	0.134856i
$124$	0	0
$125$	− 280.225i	− 0.200513i
$126$	0	0
$127$	194.980	0.136234	0.0681169	−	0.997677i	$-0.478301\pi$
$127$	194.980	0.136234	0.0681169	+	0.997677i	$0.478301\pi$
$128$	0	0
$129$	849.506	0.579805
$130$	0	0
$131$	1950.59i	1.30094i	0.759531	+	0.650471i	$0.225428\pi$
$131$	1950.59i	1.30094i	−0.759531	+	0.650471i	$0.774572\pi$
$132$	0	0
$133$	611.723i	0.398820i
$134$	0	0
$135$	30.4187	0.0193928
$136$	0	0
$137$	−568.831	−0.354733	−0.177367	−	0.984145i	$-0.556758\pi$
$137$	−568.831	−0.354733	−0.177367	+	0.984145i	$0.556758\pi$
$138$	0	0
$139$	1184.41i	0.722734i	0.932424	+	0.361367i	$0.117690\pi$
$139$	1184.41i	0.722734i	−0.932424	+	0.361367i	$0.882310\pi$
$140$	0	0
$141$	900.002i	0.537545i
$142$	0	0
$143$	197.621	0.115565
$144$	0	0
$145$	−200.720	−0.114958
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	− 2734.25i	− 1.50335i	−0.659535	−	0.751674i	$-0.729247\pi$
$149$	− 2734.25i	− 1.50335i	0.659535	−	0.751674i	$-0.270753\pi$
$150$	0	0
$151$	−267.150	−0.143976	−0.0719878	−	0.997406i	$-0.522934\pi$
$151$	−267.150	−0.143976	−0.0719878	+	0.997406i	$0.522934\pi$
$152$	0	0
$153$	623.279	0.329341
$154$	0	0
$155$	165.871i	0.0859552i
$156$	0	0
$157$	− 670.917i	− 0.341051i	−0.985353	−	0.170526i	$-0.945453\pi$
$157$	− 670.917i	− 0.341051i	0.985353	−	0.170526i	$-0.0545465\pi$
$158$	0	0
$159$	−133.311	−0.0664923
$160$	0	0
$161$	−346.933	−0.169827
$162$	0	0
$163$	− 558.841i	− 0.268539i	−0.990945	−	0.134269i	$-0.957131\pi$
$163$	− 558.841i	− 0.268539i	0.990945	−	0.134269i	$-0.0428687\pi$
$164$	0	0
$165$	− 57.2673i	− 0.0270197i
$166$	0	0
$167$	−3725.78	−1.72641	−0.863203	−	0.504857i	$-0.831545\pi$
$167$	−3725.78	−1.72641	−0.863203	+	0.504857i	$0.831545\pi$
$168$	0	0
$169$	2060.97	0.938082
$170$	0	0
$171$	786.500i	0.351726i
$172$	0	0
$173$	1574.58i	0.691985i	0.938237	+	0.345992i	$0.112458\pi$
$173$	1574.58i	0.691985i	−0.938237	+	0.345992i	$0.887542\pi$
$174$	0	0
$175$	−866.115	−0.374127
$176$	0	0
$177$	780.055	0.331257
$178$	0	0
$179$	− 302.510i	− 0.126317i	−0.998004	−	0.0631584i	$-0.979883\pi$
$179$	− 302.510i	− 0.126317i	0.998004	−	0.0631584i	$-0.0201173\pi$
$180$	0	0
$181$	2884.21i	1.18443i	0.805781	+	0.592214i	$0.201746\pi$
$181$	2884.21i	1.18443i	−0.805781	+	0.592214i	$0.798254\pi$
$182$	0	0
$183$	−453.830	−0.183323
$184$	0	0
$185$	271.646	0.107956
$186$	0	0
$187$	− 1173.40i	− 0.458865i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	−1173.79	−0.444671	−0.222336	−	0.974970i	$-0.571368\pi$
$191$	−1173.79	−0.444671	−0.222336	+	0.974970i	$0.571368\pi$
$192$	0	0
$193$	4442.36	1.65683	0.828416	−	0.560114i	$-0.189243\pi$
$193$	4442.36	1.65683	0.828416	+	0.560114i	$0.189243\pi$
$194$	0	0
$195$	39.4206i	0.0144768i
$196$	0	0
$197$	3428.87i	1.24009i	0.784568	+	0.620043i	$0.212885\pi$
$197$	3428.87i	1.24009i	−0.784568	+	0.620043i	$0.787115\pi$
$198$	0	0
$199$	2217.79	0.790025	0.395013	−	0.918676i	$-0.370740\pi$
$199$	2217.79	0.790025	0.395013	+	0.918676i	$0.370740\pi$
$200$	0	0
$201$	1905.56	0.668694
$202$	0	0
$203$	1247.13i	0.431188i
$204$	0	0
$205$	69.0852i	0.0235372i
$206$	0	0
$207$	−446.057	−0.149773
$208$	0	0
$209$	1480.69	0.490055
$210$	0	0
$211$	2952.29i	0.963243i	0.876379	+	0.481621i	$0.159952\pi$
$211$	2952.29i	0.963243i	−0.876379	+	0.481621i	$0.840048\pi$
$212$	0	0
$213$	153.856i	0.0494933i
$214$	0	0
$215$	319.024	0.101196
$216$	0	0
$217$	1030.60	0.322404
$218$	0	0
$219$	1040.93i	0.321185i
$220$	0	0
$221$	807.727i	0.245853i
$222$	0	0
$223$	534.686	0.160561	0.0802807	−	0.996772i	$-0.474418\pi$
$223$	534.686	0.160561	0.0802807	+	0.996772i	$0.474418\pi$
$224$	0	0
$225$	−1113.58	−0.329949
$226$	0	0
$227$	4975.24i	1.45471i	0.686263	+	0.727353i	$0.259250\pi$
$227$	4975.24i	1.45471i	−0.686263	+	0.727353i	$0.740750\pi$
$228$	0	0
$229$	4084.45i	1.17864i	0.807901	+	0.589318i	$0.200604\pi$
$229$	4084.45i	1.17864i	−0.807901	+	0.589318i	$0.799396\pi$
$230$	0	0
$231$	−355.817	−0.101347
$232$	0	0
$233$	5277.91	1.48398	0.741990	−	0.670411i	$-0.233882\pi$
$233$	5277.91	1.48398	0.741990	+	0.670411i	$0.233882\pi$
$234$	0	0
$235$	337.987i	0.0938206i
$236$	0	0
$237$	− 521.593i	− 0.142958i
$238$	0	0
$239$	−1005.96	−0.272260	−0.136130	−	0.990691i	$-0.543466\pi$
$239$	−1005.96	−0.272260	−0.136130	+	0.990691i	$0.543466\pi$
$240$	0	0
$241$	4550.82	1.21636	0.608182	−	0.793797i	$-0.291899\pi$
$241$	4550.82	1.21636	0.608182	+	0.793797i	$0.291899\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 55.2044i	− 0.0143954i
$246$	0	0
$247$	−1019.25	−0.262564
$248$	0	0
$249$	1572.27	0.400154
$250$	0	0
$251$	54.3052i	0.0136562i	0.999977	+	0.00682812i	$0.00217347\pi$
$251$	54.3052i	0.0136562i	−0.999977	+	0.00682812i	$0.997827\pi$
$252$	0	0
$253$	839.761i	0.208677i
$254$	0	0
$255$	234.066	0.0574816
$256$	0	0
$257$	−2807.29	−0.681378	−0.340689	−	0.940176i	$-0.610660\pi$
$257$	−2807.29	−0.681378	−0.340689	+	0.940176i	$0.610660\pi$
$258$	0	0
$259$	− 1687.81i	− 0.404924i
$260$	0	0
$261$	1603.45i	0.380272i
$262$	0	0
$263$	−2569.52	−0.602447	−0.301223	−	0.953554i	$-0.597395\pi$
$263$	−2569.52	−0.602447	−0.301223	+	0.953554i	$0.597395\pi$
$264$	0	0
$265$	−50.0638	−0.0116053
$266$	0	0
$267$	− 1799.21i	− 0.412396i
$268$	0	0
$269$	− 960.698i	− 0.217750i	−0.994055	−	0.108875i	$-0.965275\pi$
$269$	− 960.698i	− 0.217750i	0.994055	−	0.108875i	$-0.0347248\pi$
$270$	0	0
$271$	−2663.32	−0.596992	−0.298496	−	0.954411i	$-0.596485\pi$
$271$	−2663.32	−0.596992	−0.298496	+	0.954411i	$0.596485\pi$
$272$	0	0
$273$	244.931	0.0543000
$274$	0	0
$275$	2096.45i	0.459712i
$276$	0	0
$277$	195.515i	0.0424091i	0.999775	+	0.0212046i	$0.00675013\pi$
$277$	195.515i	0.0424091i	−0.999775	+	0.0212046i	$0.993250\pi$
$278$	0	0
$279$	1325.06	0.284334
$280$	0	0
$281$	−5988.66	−1.27137	−0.635683	−	0.771951i	$-0.719281\pi$
$281$	−5988.66	−1.27137	−0.635683	+	0.771951i	$0.719281\pi$
$282$	0	0
$283$	2946.99i	0.619013i	0.950897	+	0.309506i	$0.100164\pi$
$283$	2946.99i	0.619013i	−0.950897	+	0.309506i	$0.899836\pi$
$284$	0	0
$285$	295.362i	0.0613886i
$286$	0	0
$287$	429.245	0.0882841
$288$	0	0
$289$	−116.990	−0.0238124
$290$	0	0
$291$	− 1406.97i	− 0.283429i
$292$	0	0
$293$	901.308i	0.179710i	0.995955	+	0.0898549i	$0.0286403\pi$
$293$	901.308i	0.179710i	−0.995955	+	0.0898549i	$0.971360\pi$
$294$	0	0
$295$	292.942	0.0578161
$296$	0	0
$297$	−457.479	−0.0893793
$298$	0	0
$299$	− 578.059i	− 0.111806i
$300$	0	0
$301$	− 1982.18i	− 0.379571i
$302$	0	0
$303$	4848.29	0.919231
$304$	0	0
$305$	−170.431	−0.0319963
$306$	0	0
$307$	− 1421.75i	− 0.264312i	−0.991229	−	0.132156i	$-0.957810\pi$
$307$	− 1421.75i	− 0.264312i	0.991229	−	0.132156i	$-0.0421900\pi$
$308$	0	0
$309$	4974.60i	0.915841i
$310$	0	0
$311$	−4673.63	−0.852146	−0.426073	−	0.904689i	$-0.640103\pi$
$311$	−4673.63	−0.852146	−0.426073	+	0.904689i	$0.640103\pi$
$312$	0	0
$313$	2600.53	0.469619	0.234810	−	0.972041i	$-0.424553\pi$
$313$	2600.53	0.469619	0.234810	+	0.972041i	$0.424553\pi$
$314$	0	0
$315$	− 70.9771i	− 0.0126956i
$316$	0	0
$317$	− 2766.95i	− 0.490244i	−0.969492	−	0.245122i	$-0.921172\pi$
$317$	− 2766.95i	− 0.490244i	0.969492	−	0.245122i	$-0.0788280\pi$
$318$	0	0
$319$	3018.70	0.529828
$320$	0	0
$321$	3230.52	0.561713
$322$	0	0
$323$	6051.97i	1.04254i
$324$	0	0
$325$	− 1443.12i	− 0.246307i
$326$	0	0
$327$	428.249	0.0724227
$328$	0	0
$329$	2100.01	0.351906
$330$	0	0
$331$	3545.04i	0.588679i	0.955701	+	0.294340i	$0.0950997\pi$
$331$	3545.04i	0.588679i	−0.955701	+	0.294340i	$0.904900\pi$
$332$	0	0
$333$	− 2170.04i	− 0.357110i
$334$	0	0
$335$	715.613	0.116711
$336$	0	0
$337$	−1357.19	−0.219379	−0.109689	−	0.993966i	$-0.534986\pi$
$337$	−1357.19	−0.219379	−0.109689	+	0.993966i	$0.534986\pi$
$338$	0	0
$339$	− 4194.56i	− 0.672028i
$340$	0	0
$341$	− 2494.59i	− 0.396158i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−167.512	−0.0261408
$346$	0	0
$347$	10543.8i	1.63119i	0.578626	+	0.815593i	$0.303589\pi$
$347$	10543.8i	1.63119i	−0.578626	+	0.815593i	$0.696411\pi$
$348$	0	0
$349$	2715.58i	0.416509i	0.978075	+	0.208254i	$0.0667782\pi$
$349$	2715.58i	0.416509i	−0.978075	+	0.208254i	$0.933222\pi$
$350$	0	0
$351$	314.911	0.0478881
$352$	0	0
$353$	−3417.09	−0.515223	−0.257611	−	0.966249i	$-0.582935\pi$
$353$	−3417.09	−0.515223	−0.257611	+	0.966249i	$0.582935\pi$
$354$	0	0
$355$	57.7793i	0.00863832i
$356$	0	0
$357$	− 1454.32i	− 0.215604i
$358$	0	0
$359$	−6856.79	−1.00804	−0.504021	−	0.863691i	$-0.668147\pi$
$359$	−6856.79	−1.00804	−0.504021	+	0.863691i	$0.668147\pi$
$360$	0	0
$361$	−777.826	−0.113402
$362$	0	0
$363$	− 3131.74i	− 0.452819i
$364$	0	0
$365$	390.910i	0.0560580i
$366$	0	0
$367$	3759.36	0.534706	0.267353	−	0.963599i	$-0.413851\pi$
$367$	3759.36	0.534706	0.267353	+	0.963599i	$0.413851\pi$
$368$	0	0
$369$	551.887	0.0778593
$370$	0	0
$371$	311.060i	0.0435295i
$372$	0	0
$373$	1961.48i	0.272283i	0.990689	+	0.136142i	$0.0434702\pi$
$373$	1961.48i	0.272283i	−0.990689	+	0.136142i	$0.956530\pi$
$374$	0	0
$375$	−840.675	−0.115766
$376$	0	0
$377$	−2077.96	−0.283874
$378$	0	0
$379$	− 4659.88i	− 0.631562i	−0.948832	−	0.315781i	$-0.897733\pi$
$379$	− 4659.88i	− 0.631562i	0.948832	−	0.315781i	$-0.102267\pi$
$380$	0	0
$381$	− 584.940i	− 0.0786546i
$382$	0	0
$383$	−6561.15	−0.875351	−0.437675	−	0.899133i	$-0.644198\pi$
$383$	−6561.15	−0.875351	−0.437675	+	0.899133i	$0.644198\pi$
$384$	0	0
$385$	−133.624	−0.0176886
$386$	0	0
$387$	− 2548.52i	− 0.334751i
$388$	0	0
$389$	− 6155.68i	− 0.802328i	−0.916006	−	0.401164i	$-0.868606\pi$
$389$	− 6155.68i	− 0.802328i	0.916006	−	0.401164i	$-0.131394\pi$
$390$	0	0
$391$	−3432.32	−0.443939
$392$	0	0
$393$	5851.76	0.751099
$394$	0	0
$395$	− 195.879i	− 0.0249512i
$396$	0	0
$397$	− 829.907i	− 0.104916i	−0.998623	−	0.0524582i	$-0.983294\pi$
$397$	− 829.907i	− 0.104916i	0.998623	−	0.0524582i	$-0.0167056\pi$
$398$	0	0
$399$	1835.17	0.230259
$400$	0	0
$401$	−12733.5	−1.58574	−0.792871	−	0.609389i	$-0.791415\pi$
$401$	−12733.5	−1.58574	−0.792871	+	0.609389i	$0.791415\pi$
$402$	0	0
$403$	1717.18i	0.212256i
$404$	0	0
$405$	− 91.2562i	− 0.0111964i
$406$	0	0
$407$	−4085.39	−0.497555
$408$	0	0
$409$	−372.265	−0.0450057	−0.0225028	−	0.999747i	$-0.507163\pi$
$409$	−372.265	−0.0450057	−0.0225028	+	0.999747i	$0.507163\pi$
$410$	0	0
$411$	1706.49i	0.204805i
$412$	0	0
$413$	− 1820.13i	− 0.216859i
$414$	0	0
$415$	590.450	0.0698411
$416$	0	0
$417$	3553.22	0.417271
$418$	0	0
$419$	3440.17i	0.401106i	0.979683	+	0.200553i	$0.0642739\pi$
$419$	3440.17i	0.401106i	−0.979683	+	0.200553i	$0.935726\pi$
$420$	0	0
$421$	2378.30i	0.275323i	0.990479	+	0.137662i	$0.0439587\pi$
$421$	2378.30i	0.275323i	−0.990479	+	0.137662i	$0.956041\pi$
$422$	0	0
$423$	2700.01	0.310352
$424$	0	0
$425$	−8568.75	−0.977990
$426$	0	0
$427$	1058.94i	0.120013i
$428$	0	0
$429$	− 592.862i	− 0.0667218i
$430$	0	0
$431$	13684.1	1.52933	0.764666	−	0.644427i	$-0.222904\pi$
$431$	13684.1	1.52933	0.764666	+	0.644427i	$0.222904\pi$
$432$	0	0
$433$	−1549.30	−0.171950	−0.0859751	−	0.996297i	$-0.527401\pi$
$433$	−1549.30	−0.171950	−0.0859751	+	0.996297i	$0.527401\pi$
$434$	0	0
$435$	602.160i	0.0663709i
$436$	0	0
$437$	− 4331.16i	− 0.474113i
$438$	0	0
$439$	−16525.0	−1.79657	−0.898284	−	0.439415i	$-0.855186\pi$
$439$	−16525.0	−1.79657	−0.898284	+	0.439415i	$0.855186\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	1376.91i	0.147673i	0.997270	+	0.0738363i	$0.0235242\pi$
$443$	1376.91i	0.147673i	−0.997270	+	0.0738363i	$0.976476\pi$
$444$	0	0
$445$	− 675.674i	− 0.0719776i
$446$	0	0
$447$	−8202.76	−0.867958
$448$	0	0
$449$	−7445.74	−0.782597	−0.391298	−	0.920264i	$-0.627974\pi$
$449$	−7445.74	−0.782597	−0.391298	+	0.920264i	$0.627974\pi$
$450$	0	0
$451$	− 1039.00i	− 0.108480i
$452$	0	0
$453$	801.449i	0.0831244i
$454$	0	0
$455$	91.9814	0.00947727
$456$	0	0
$457$	4003.36	0.409780	0.204890	−	0.978785i	$-0.434316\pi$
$457$	4003.36	0.409780	0.204890	+	0.978785i	$0.434316\pi$
$458$	0	0
$459$	− 1869.84i	− 0.190145i
$460$	0	0
$461$	− 15246.2i	− 1.54032i	−0.637852	−	0.770159i	$-0.720177\pi$
$461$	− 15246.2i	− 1.54032i	0.637852	−	0.770159i	$-0.279823\pi$
$462$	0	0
$463$	16524.8	1.65869	0.829343	−	0.558740i	$-0.188715\pi$
$463$	16524.8	1.65869	0.829343	+	0.558740i	$0.188715\pi$
$464$	0	0
$465$	497.612	0.0496263
$466$	0	0
$467$	− 6013.04i	− 0.595825i	−0.954593	−	0.297912i	$-0.903710\pi$
$467$	− 6013.04i	− 0.595825i	0.954593	−	0.297912i	$-0.0962903\pi$
$468$	0	0
$469$	− 4446.30i	− 0.437763i
$470$	0	0
$471$	−2012.75	−0.196906
$472$	0	0
$473$	−4797.92	−0.466403
$474$	0	0
$475$	− 10812.7i	− 1.04446i
$476$	0	0
$477$	399.934i	0.0383894i
$478$	0	0
$479$	8271.63	0.789020	0.394510	−	0.918892i	$-0.370914\pi$
$479$	8271.63	0.789020	0.394510	+	0.918892i	$0.370914\pi$
$480$	0	0
$481$	2812.22	0.266583
$482$	0	0
$483$	1040.80i	0.0980498i
$484$	0	0
$485$	− 528.372i	− 0.0494683i
$486$	0	0
$487$	15872.6	1.47691	0.738456	−	0.674301i	$-0.235555\pi$
$487$	15872.6	1.47691	0.738456	+	0.674301i	$0.235555\pi$
$488$	0	0
$489$	−1676.52	−0.155041
$490$	0	0
$491$	5409.77i	0.497229i	0.968602	+	0.248615i	$0.0799752\pi$
$491$	5409.77i	0.497229i	−0.968602	+	0.248615i	$0.920025\pi$
$492$	0	0
$493$	12338.2i	1.12715i
$494$	0	0
$495$	−171.802	−0.0155998
$496$	0	0
$497$	358.998	0.0324010
$498$	0	0
$499$	− 13151.8i	− 1.17987i	−0.807449	−	0.589937i	$-0.799153\pi$
$499$	− 13151.8i	− 1.17987i	0.807449	−	0.589937i	$-0.200847\pi$
$500$	0	0
$501$	11177.4i	0.996741i
$502$	0	0
$503$	−10782.7	−0.955815	−0.477907	−	0.878410i	$-0.658605\pi$
$503$	−10782.7	−0.955815	−0.477907	+	0.878410i	$0.658605\pi$
$504$	0	0
$505$	1820.73	0.160438
$506$	0	0
$507$	− 6182.90i	− 0.541602i
$508$	0	0
$509$	− 1792.29i	− 0.156074i	−0.996950	−	0.0780371i	$-0.975135\pi$
$509$	− 1792.29i	− 0.156074i	0.996950	−	0.0780371i	$-0.0248653\pi$
$510$	0	0
$511$	2428.83	0.210265
$512$	0	0
$513$	2359.50	0.203069
$514$	0	0
$515$	1868.16i	0.159847i
$516$	0	0
$517$	− 5083.12i	− 0.432409i
$518$	0	0
$519$	4723.75	0.399518
$520$	0	0
$521$	−16534.6	−1.39039	−0.695197	−	0.718819i	$-0.744683\pi$
$521$	−16534.6	−1.39039	−0.695197	+	0.718819i	$0.744683\pi$
$522$	0	0
$523$	− 14600.5i	− 1.22072i	−0.792125	−	0.610359i	$-0.791025\pi$
$523$	− 14600.5i	− 1.22072i	0.792125	−	0.610359i	$-0.208975\pi$
$524$	0	0
$525$	2598.35i	0.216002i
$526$	0	0
$527$	10196.1	0.842784
$528$	0	0
$529$	−9710.62	−0.798111
$530$	0	0
$531$	− 2340.17i	− 0.191252i
$532$	0	0
$533$	715.207i	0.0581221i
$534$	0	0
$535$	1213.19	0.0980387
$536$	0	0
$537$	−907.531	−0.0729290
$538$	0	0
$539$	830.240i	0.0663469i
$540$	0	0
$541$	− 17730.8i	− 1.40907i	−0.709670	−	0.704535i	$-0.751156\pi$
$541$	− 17730.8i	− 1.40907i	0.709670	−	0.704535i	$-0.248844\pi$
$542$	0	0
$543$	8652.63	0.683830
$544$	0	0
$545$	160.825	0.0126403
$546$	0	0
$547$	− 13764.9i	− 1.07595i	−0.842962	−	0.537973i	$-0.819190\pi$
$547$	− 13764.9i	− 1.07595i	0.842962	−	0.537973i	$-0.180810\pi$
$548$	0	0
$549$	1361.49i	0.105842i
$550$	0	0
$551$	−15569.3	−1.20377
$552$	0	0
$553$	−1217.05	−0.0935881
$554$	0	0
$555$	− 814.937i	− 0.0623282i
$556$	0	0
$557$	− 8864.33i	− 0.674315i	−0.941448	−	0.337158i	$-0.890534\pi$
$557$	− 8864.33i	− 0.674315i	0.941448	−	0.337158i	$-0.109466\pi$
$558$	0	0
$559$	3302.70	0.249892
$560$	0	0
$561$	−3520.21	−0.264926
$562$	0	0
$563$	11425.6i	0.855293i	0.903946	+	0.427647i	$0.140657\pi$
$563$	11425.6i	0.855293i	−0.903946	+	0.427647i	$0.859343\pi$
$564$	0	0
$565$	− 1575.23i	− 0.117293i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−6486.88	−0.477934	−0.238967	−	0.971028i	$-0.576809\pi$
$569$	−6486.88	−0.477934	−0.238967	+	0.971028i	$0.576809\pi$
$570$	0	0
$571$	− 11217.7i	− 0.822148i	−0.911602	−	0.411074i	$-0.865154\pi$
$571$	− 11217.7i	− 0.822148i	0.911602	−	0.411074i	$-0.134846\pi$
$572$	0	0
$573$	3521.36i	0.256731i
$574$	0	0
$575$	6132.33	0.444758
$576$	0	0
$577$	−2519.81	−0.181804	−0.0909022	−	0.995860i	$-0.528975\pi$
$577$	−2519.81	−0.181804	−0.0909022	+	0.995860i	$0.528975\pi$
$578$	0	0
$579$	− 13327.1i	− 0.956572i
$580$	0	0
$581$	− 3668.63i	− 0.261963i
$582$	0	0
$583$	752.929	0.0534873
$584$	0	0
$585$	118.262	0.00835816
$586$	0	0
$587$	2346.94i	0.165023i	0.996590	+	0.0825116i	$0.0262942\pi$
$587$	2346.94i	0.165023i	−0.996590	+	0.0825116i	$0.973706\pi$
$588$	0	0
$589$	12866.2i	0.900069i
$590$	0	0
$591$	10286.6	0.715964
$592$	0	0
$593$	−6093.83	−0.421996	−0.210998	−	0.977487i	$-0.567671\pi$
$593$	−6093.83	−0.421996	−0.210998	+	0.977487i	$0.567671\pi$
$594$	0	0
$595$	− 546.155i	− 0.0376305i
$596$	0	0
$597$	− 6653.37i	− 0.456121i
$598$	0	0
$599$	−6469.67	−0.441308	−0.220654	−	0.975352i	$-0.570819\pi$
$599$	−6469.67	−0.441308	−0.220654	+	0.975352i	$0.570819\pi$
$600$	0	0
$601$	−4784.14	−0.324708	−0.162354	−	0.986733i	$-0.551909\pi$
$601$	−4784.14	−0.324708	−0.162354	+	0.986733i	$0.551909\pi$
$602$	0	0
$603$	− 5716.67i	− 0.386071i
$604$	0	0
$605$	− 1176.09i	− 0.0790330i
$606$	0	0
$607$	26120.8	1.74664	0.873319	−	0.487148i	$-0.161963\pi$
$607$	26120.8	1.74664	0.873319	+	0.487148i	$0.161963\pi$
$608$	0	0
$609$	3741.38	0.248947
$610$	0	0
$611$	3499.02i	0.231678i
$612$	0	0
$613$	11205.0i	0.738279i	0.929374	+	0.369140i	$0.120348\pi$
$613$	11205.0i	0.738279i	−0.929374	+	0.369140i	$0.879652\pi$
$614$	0	0
$615$	207.256	0.0135892
$616$	0	0
$617$	−21325.0	−1.39143	−0.695715	−	0.718318i	$-0.744912\pi$
$617$	−21325.0	−1.39143	−0.695715	+	0.718318i	$0.744912\pi$
$618$	0	0
$619$	5654.26i	0.367147i	0.983006	+	0.183573i	$0.0587665\pi$
$619$	5654.26i	0.367147i	−0.983006	+	0.183573i	$0.941233\pi$
$620$	0	0
$621$	1338.17i	0.0864718i
$622$	0	0
$623$	−4198.15	−0.269976
$624$	0	0
$625$	15150.6	0.969641
$626$	0	0
$627$	− 4442.07i	− 0.282933i
$628$	0	0
$629$	− 16698.0i	− 1.05850i
$630$	0	0
$631$	−15081.9	−0.951508	−0.475754	−	0.879578i	$-0.657825\pi$
$631$	−15081.9	−0.951508	−0.475754	+	0.879578i	$0.657825\pi$
$632$	0	0
$633$	8856.88	0.556129
$634$	0	0
$635$	− 219.669i	− 0.0137280i
$636$	0	0
$637$	− 571.506i	− 0.0355477i
$638$	0	0
$639$	461.569	0.0285750
$640$	0	0
$641$	−12856.9	−0.792223	−0.396112	−	0.918202i	$-0.629641\pi$
$641$	−12856.9	−0.792223	−0.396112	+	0.918202i	$0.629641\pi$
$642$	0	0
$643$	− 17420.3i	− 1.06841i	−0.845355	−	0.534205i	$-0.820611\pi$
$643$	− 17420.3i	− 1.06841i	0.845355	−	0.534205i	$-0.179389\pi$
$644$	0	0
$645$	− 957.071i	− 0.0584258i
$646$	0	0
$647$	1673.67	0.101698	0.0508492	−	0.998706i	$-0.483807\pi$
$647$	1673.67	0.101698	0.0508492	+	0.998706i	$0.483807\pi$
$648$	0	0
$649$	−4405.67	−0.266468
$650$	0	0
$651$	− 3091.80i	− 0.186140i
$652$	0	0
$653$	5827.22i	0.349214i	0.984638	+	0.174607i	$0.0558655\pi$
$653$	5827.22i	0.349214i	−0.984638	+	0.174607i	$0.944134\pi$
$654$	0	0
$655$	2197.57	0.131093
$656$	0	0
$657$	3122.79	0.185436
$658$	0	0
$659$	6512.91i	0.384988i	0.981298	+	0.192494i	$0.0616575\pi$
$659$	6512.91i	0.384988i	−0.981298	+	0.192494i	$0.938342\pi$
$660$	0	0
$661$	− 5536.34i	− 0.325777i	−0.986644	−	0.162889i	$-0.947919\pi$
$661$	− 5536.34i	− 0.325777i	0.986644	−	0.162889i	$-0.0520811\pi$
$662$	0	0
$663$	2423.18	0.141944
$664$	0	0
$665$	689.179	0.0401883
$666$	0	0
$667$	− 8830.00i	− 0.512592i
$668$	0	0
$669$	− 1604.06i	− 0.0927002i
$670$	0	0
$671$	2563.18	0.147467
$672$	0	0
$673$	−27240.2	−1.56023	−0.780114	−	0.625637i	$-0.784839\pi$
$673$	−27240.2	−1.56023	−0.780114	+	0.625637i	$0.784839\pi$
$674$	0	0
$675$	3340.73i	0.190496i
$676$	0	0
$677$	− 9340.54i	− 0.530260i	−0.964213	−	0.265130i	$-0.914585\pi$
$677$	− 9340.54i	− 0.530260i	0.964213	−	0.265130i	$-0.0854149\pi$
$678$	0	0
$679$	−3282.92	−0.185548
$680$	0	0
$681$	14925.7	0.839875
$682$	0	0
$683$	− 7644.87i	− 0.428291i	−0.976802	−	0.214145i	$-0.931303\pi$
$683$	− 7644.87i	− 0.428291i	0.976802	−	0.214145i	$-0.0686966\pi$
$684$	0	0
$685$	640.856i	0.0357458i
$686$	0	0
$687$	12253.3	0.680486
$688$	0	0
$689$	−518.287	−0.0286577
$690$	0	0
$691$	− 21772.5i	− 1.19865i	−0.800507	−	0.599323i	$-0.795436\pi$
$691$	− 21772.5i	− 1.19865i	0.800507	−	0.599323i	$-0.204564\pi$
$692$	0	0
$693$	1067.45i	0.0585125i
$694$	0	0
$695$	1334.38	0.0728284
$696$	0	0
$697$	4246.66	0.230780
$698$	0	0
$699$	− 15833.7i	− 0.856776i
$700$	0	0
$701$	17728.0i	0.955176i	0.878584	+	0.477588i	$0.158489\pi$
$701$	17728.0i	0.955176i	−0.878584	+	0.477588i	$0.841511\pi$
$702$	0	0
$703$	21070.8	1.13044
$704$	0	0
$705$	1013.96	0.0541674
$706$	0	0
$707$	− 11312.7i	− 0.601778i
$708$	0	0
$709$	− 17545.2i	− 0.929372i	−0.885475	−	0.464686i	$-0.846167\pi$
$709$	− 17545.2i	− 0.929372i	0.885475	−	0.464686i	$-0.153833\pi$
$710$	0	0
$711$	−1564.78	−0.0825369
$712$	0	0
$713$	−7296.93	−0.383271
$714$	0	0
$715$	− 222.643i	− 0.0116453i
$716$	0	0
$717$	3017.87i	0.157189i
$718$	0	0
$719$	−20382.2	−1.05720	−0.528600	−	0.848871i	$-0.677283\pi$
$719$	−20382.2	−1.05720	−0.528600	+	0.848871i	$0.677283\pi$
$720$	0	0
$721$	11607.4	0.599559
$722$	0	0
$723$	− 13652.4i	− 0.702268i
$724$	0	0
$725$	− 22044.0i	− 1.12923i
$726$	0	0
$727$	−5123.30	−0.261366	−0.130683	−	0.991424i	$-0.541717\pi$
$727$	−5123.30	−0.261366	−0.130683	+	0.991424i	$0.541717\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 19610.3i	− 0.992223i
$732$	0	0
$733$	− 11859.4i	− 0.597595i	−0.954317	−	0.298798i	$-0.903414\pi$
$733$	− 11859.4i	− 0.597595i	0.954317	−	0.298798i	$-0.0965855\pi$
$734$	0	0
$735$	−165.613	−0.00831120
$736$	0	0
$737$	−10762.4	−0.537907
$738$	0	0
$739$	24053.3i	1.19731i	0.801006	+	0.598656i	$0.204298\pi$
$739$	24053.3i	1.19731i	−0.801006	+	0.598656i	$0.795702\pi$
$740$	0	0
$741$	3057.75i	0.151592i
$742$	0	0
$743$	29417.3	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$743$	29417.3	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$744$	0	0
$745$	−3080.47	−0.151489
$746$	0	0
$747$	− 4716.80i	− 0.231029i
$748$	0	0
$749$	− 7537.87i	− 0.367727i
$750$	0	0
$751$	16065.6	0.780613	0.390306	−	0.920685i	$-0.372369\pi$
$751$	16065.6	0.780613	0.390306	+	0.920685i	$0.372369\pi$
$752$	0	0
$753$	162.916	0.00788443
$754$	0	0
$755$	300.976i	0.0145081i
$756$	0	0
$757$	34975.4i	1.67926i	0.543157	+	0.839631i	$0.317229\pi$
$757$	34975.4i	1.67926i	−0.543157	+	0.839631i	$0.682771\pi$
$758$	0	0
$759$	2519.28	0.120480
$760$	0	0
$761$	−2817.64	−0.134217	−0.0671087	−	0.997746i	$-0.521377\pi$
$761$	−2817.64	−0.134217	−0.0671087	+	0.997746i	$0.521377\pi$
$762$	0	0
$763$	− 999.247i	− 0.0474118i
$764$	0	0
$765$	− 702.199i	− 0.0331870i
$766$	0	0
$767$	3032.70	0.142770
$768$	0	0
$769$	25638.1	1.20225	0.601126	−	0.799154i	$-0.294719\pi$
$769$	25638.1	1.20225	0.601126	+	0.799154i	$0.294719\pi$
$770$	0	0
$771$	8421.88i	0.393394i
$772$	0	0
$773$	− 8918.71i	− 0.414985i	−0.978237	−	0.207493i	$-0.933470\pi$
$773$	− 8918.71i	− 0.414985i	0.978237	−	0.207493i	$-0.0665303\pi$
$774$	0	0
$775$	−18216.7	−0.844340
$776$	0	0
$777$	−5063.43	−0.233783
$778$	0	0
$779$	5358.75i	0.246466i
$780$	0	0
$781$	− 868.965i	− 0.0398131i
$782$	0	0
$783$	4810.35	0.219550
$784$	0	0
$785$	−755.869	−0.0343670
$786$	0	0
$787$	− 20743.3i	− 0.939539i	−0.882789	−	0.469770i	$-0.844337\pi$
$787$	− 20743.3i	− 0.939539i	0.882789	−	0.469770i	$-0.155663\pi$
$788$	0	0
$789$	7708.56i	0.347823i
$790$	0	0
$791$	−9787.32	−0.439946
$792$	0	0
$793$	−1764.40	−0.0790109
$794$	0	0
$795$	150.191i	0.00670030i
$796$	0	0
$797$	14348.5i	0.637705i	0.947804	+	0.318852i	$0.103297\pi$
$797$	14348.5i	0.637705i	−0.947804	+	0.318852i	$0.896703\pi$
$798$	0	0
$799$	20776.0	0.919904
$800$	0	0
$801$	−5397.62	−0.238097
$802$	0	0
$803$	− 5879.05i	− 0.258365i
$804$	0	0
$805$	390.862i	0.0171131i
$806$	0	0
$807$	−2882.09	−0.125718
$808$	0	0
$809$	−5358.04	−0.232854	−0.116427	−	0.993199i	$-0.537144\pi$
$809$	−5358.04	−0.232854	−0.116427	+	0.993199i	$0.537144\pi$
$810$	0	0
$811$	9747.34i	0.422041i	0.977482	+	0.211020i	$0.0676787\pi$
$811$	9747.34i	0.422041i	−0.977482	+	0.211020i	$0.932321\pi$
$812$	0	0
$813$	7989.95i	0.344674i
$814$	0	0
$815$	−629.602	−0.0270601
$816$	0	0
$817$	24745.8	1.05966
$818$	0	0
$819$	− 734.793i	− 0.0313501i
$820$	0	0
$821$	− 33778.7i	− 1.43592i	−0.696087	−	0.717958i	$-0.745077\pi$
$821$	− 33778.7i	− 1.43592i	0.696087	−	0.717958i	$-0.254923\pi$
$822$	0	0
$823$	−27072.2	−1.14663	−0.573317	−	0.819334i	$-0.694343\pi$
$823$	−27072.2	−1.14663	−0.573317	+	0.819334i	$0.694343\pi$
$824$	0	0
$825$	6289.36	0.265415
$826$	0	0
$827$	32860.0i	1.38169i	0.723005	+	0.690843i	$0.242760\pi$
$827$	32860.0i	1.38169i	−0.723005	+	0.690843i	$0.757240\pi$
$828$	0	0
$829$	− 1278.56i	− 0.0535659i	−0.999641	−	0.0267829i	$-0.991474\pi$
$829$	− 1278.56i	− 0.0535659i	0.999641	−	0.0267829i	$-0.00852629\pi$
$830$	0	0
$831$	586.544	0.0244849
$832$	0	0
$833$	−3393.41	−0.141146
$834$	0	0
$835$	4197.54i	0.173966i
$836$	0	0
$837$	− 3975.17i	− 0.164160i
$838$	0	0
$839$	−18202.9	−0.749026	−0.374513	−	0.927222i	$-0.622190\pi$
$839$	−18202.9	−0.749026	−0.374513	+	0.927222i	$0.622190\pi$
$840$	0	0
$841$	−7352.37	−0.301462
$842$	0	0
$843$	17966.0i	0.734023i
$844$	0	0
$845$	− 2321.93i	− 0.0945286i
$846$	0	0
$847$	−7307.38	−0.296440
$848$	0	0
$849$	8840.98	0.357387
$850$	0	0
$851$	11950.2i	0.481370i
$852$	0	0
$853$	− 30220.2i	− 1.21304i	−0.795070	−	0.606518i	$-0.792566\pi$
$853$	− 30220.2i	− 1.21304i	0.795070	−	0.606518i	$-0.207434\pi$
$854$	0	0
$855$	886.087	0.0354427
$856$	0	0
$857$	−12720.0	−0.507011	−0.253505	−	0.967334i	$-0.581584\pi$
$857$	−12720.0	−0.507011	−0.253505	+	0.967334i	$0.581584\pi$
$858$	0	0
$859$	15021.6i	0.596660i	0.954463	+	0.298330i	$0.0964296\pi$
$859$	15021.6i	0.596660i	−0.954463	+	0.298330i	$0.903570\pi$
$860$	0	0
$861$	− 1287.74i	− 0.0509708i
$862$	0	0
$863$	−15840.4	−0.624813	−0.312407	−	0.949948i	$-0.601135\pi$
$863$	−15840.4	−0.624813	−0.312407	+	0.949948i	$0.601135\pi$
$864$	0	0
$865$	1773.96	0.0697299
$866$	0	0
$867$	350.970i	0.0137481i
$868$	0	0
$869$	2945.90i	0.114997i
$870$	0	0
$871$	7408.41	0.288203
$872$	0	0
$873$	−4220.90	−0.163638
$874$	0	0
$875$	1961.58i	0.0757867i
$876$	0	0
$877$	− 5681.32i	− 0.218751i	−0.994001	−	0.109375i	$-0.965115\pi$
$877$	− 5681.32i	− 0.218751i	0.994001	−	0.109375i	$-0.0348851\pi$
$878$	0	0
$879$	2703.92	0.103755
$880$	0	0
$881$	−51816.0	−1.98153	−0.990764	−	0.135600i	$-0.956704\pi$
$881$	−51816.0	−1.98153	−0.990764	+	0.135600i	$0.956704\pi$
$882$	0	0
$883$	23486.9i	0.895129i	0.894252	+	0.447564i	$0.147708\pi$
$883$	23486.9i	0.895129i	−0.894252	+	0.447564i	$0.852292\pi$
$884$	0	0
$885$	− 878.826i	− 0.0333801i
$886$	0	0
$887$	−34465.5	−1.30467	−0.652333	−	0.757933i	$-0.726210\pi$
$887$	−34465.5	−1.30467	−0.652333	+	0.757933i	$0.726210\pi$
$888$	0	0
$889$	−1364.86	−0.0514915
$890$	0	0
$891$	1372.44i	0.0516031i
$892$	0	0
$893$	26216.8i	0.982430i
$894$	0	0
$895$	−340.814	−0.0127287
$896$	0	0
$897$	−1734.18	−0.0645513
$898$	0	0
$899$	26230.4i	0.973118i
$900$	0	0
$901$	3077.42i	0.113789i
$902$	0	0
$903$	−5946.54	−0.219146
$904$	0	0
$905$	3249.41	0.119352
$906$	0	0
$907$	− 46435.6i	− 1.69996i	−0.526811	−	0.849982i	$-0.676613\pi$
$907$	− 46435.6i	− 1.69996i	0.526811	−	0.849982i	$-0.323387\pi$
$908$	0	0
$909$	− 14544.9i	− 0.530718i
$910$	0	0
$911$	50611.5	1.84065	0.920326	−	0.391152i	$-0.127923\pi$
$911$	50611.5	1.84065	0.920326	+	0.391152i	$0.127923\pi$
$912$	0	0
$913$	−8880.00	−0.321890
$914$	0	0
$915$	511.294i	0.0184731i
$916$	0	0
$917$	− 13654.1i	− 0.491710i
$918$	0	0
$919$	−10017.6	−0.359577	−0.179788	−	0.983705i	$-0.557541\pi$
$919$	−10017.6	−0.359577	−0.179788	+	0.983705i	$0.557541\pi$
$920$	0	0
$921$	−4265.26	−0.152601
$922$	0	0
$923$	598.162i	0.0213313i
$924$	0	0
$925$	29833.4i	1.06045i
$926$	0	0
$927$	14923.8	0.528761
$928$	0	0
$929$	−7218.71	−0.254939	−0.127469	−	0.991842i	$-0.540685\pi$
$929$	−7218.71	−0.254939	−0.127469	+	0.991842i	$0.540685\pi$
$930$	0	0
$931$	− 4282.06i	− 0.150740i
$932$	0	0
$933$	14020.9i	0.491987i
$934$	0	0
$935$	−1321.98	−0.0462389
$936$	0	0
$937$	16890.3	0.588883	0.294442	−	0.955669i	$-0.404866\pi$
$937$	16890.3	0.588883	0.294442	+	0.955669i	$0.404866\pi$
$938$	0	0
$939$	− 7801.60i	− 0.271135i
$940$	0	0
$941$	34965.2i	1.21130i	0.795731	+	0.605650i	$0.207087\pi$
$941$	34965.2i	1.21130i	−0.795731	+	0.605650i	$0.792913\pi$
$942$	0	0
$943$	−3039.17	−0.104951
$944$	0	0
$945$	−212.931	−0.00732979
$946$	0	0
$947$	3521.05i	0.120822i	0.998174	+	0.0604112i	$0.0192412\pi$
$947$	3521.05i	0.120822i	−0.998174	+	0.0604112i	$0.980759\pi$
$948$	0	0
$949$	4046.92i	0.138428i
$950$	0	0
$951$	−8300.85	−0.283043
$952$	0	0
$953$	−19019.2	−0.646475	−0.323238	−	0.946318i	$-0.604771\pi$
$953$	−19019.2	−0.646475	−0.323238	+	0.946318i	$0.604771\pi$
$954$	0	0
$955$	1322.41i	0.0448086i
$956$	0	0
$957$	− 9056.11i	− 0.305896i
$958$	0	0
$959$	3981.81	0.134077
$960$	0	0
$961$	−8114.72	−0.272388
$962$	0	0
$963$	− 9691.55i	− 0.324305i
$964$	0	0
$965$	− 5004.86i	− 0.166956i
$966$	0	0
$967$	25730.3	0.855667	0.427834	−	0.903858i	$-0.359277\pi$
$967$	25730.3	0.855667	0.427834	+	0.903858i	$0.359277\pi$
$968$	0	0
$969$	18155.9	0.601911
$970$	0	0
$971$	26692.1i	0.882173i	0.897465	+	0.441086i	$0.145407\pi$
$971$	26692.1i	0.882173i	−0.897465	+	0.441086i	$0.854593\pi$
$972$	0	0
$973$	− 8290.84i	− 0.273168i
$974$	0	0
$975$	−4329.36	−0.142206
$976$	0	0
$977$	6865.80	0.224828	0.112414	−	0.993661i	$-0.464142\pi$
$977$	6865.80	0.224828	0.112414	+	0.993661i	$0.464142\pi$
$978$	0	0
$979$	10161.7i	0.331737i
$980$	0	0
$981$	− 1284.75i	− 0.0418133i
$982$	0	0
$983$	−36526.1	−1.18515	−0.592574	−	0.805516i	$-0.701888\pi$
$983$	−36526.1	−1.18515	−0.592574	+	0.805516i	$0.701888\pi$
$984$	0	0
$985$	3863.04	0.124961
$986$	0	0
$987$	− 6300.02i	− 0.203173i
$988$	0	0
$989$	14034.4i	0.451231i
$990$	0	0
$991$	25643.4	0.821988	0.410994	−	0.911638i	$-0.365182\pi$
$991$	25643.4	0.821988	0.410994	+	0.911638i	$0.365182\pi$
$992$	0	0
$993$	10635.1	0.339874
$994$	0	0
$995$	− 2498.61i	− 0.0796093i
$996$	0	0
$997$	58534.0i	1.85937i	0.368357	+	0.929684i	$0.379920\pi$
$997$	58534.0i	1.85937i	−0.368357	+	0.929684i	$0.620080\pi$
$998$	0	0
$999$	−6510.12	−0.206177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.4	✓	12
4.3	odd	2		1344.4.c.g.673.10	yes	12
8.3	odd	2		1344.4.c.g.673.3	yes	12
8.5	even	2	inner	1344.4.c.f.673.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.4	✓	12	1.1	even	1	trivial
1344.4.c.f.673.9	yes	12	8.5	even	2	inner
1344.4.c.g.673.3	yes	12	8.3	odd	2
1344.4.c.g.673.10	yes	12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -1.12662i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -1.12662i q^{5} -7.00000 q^{7} -9.00000 q^{9} +16.9437i q^{11} -11.6634i q^{13} -3.37986 q^{15} -69.2532 q^{17} -87.3889i q^{19} +21.0000i q^{21} +49.5619 q^{23} +123.731 q^{25} +27.0000i q^{27} -178.161i q^{29} -147.229 q^{31} +50.8310 q^{33} +7.88634i q^{35} +241.116i q^{37} -34.9901 q^{39} -61.3207 q^{41} +283.169i q^{43} +10.1396i q^{45} -300.001 q^{47} +49.0000 q^{49} +207.760i q^{51} -44.4371i q^{53} +19.0891 q^{55} -262.167 q^{57} +260.018i q^{59} -151.277i q^{61} +63.0000 q^{63} -13.1402 q^{65} +635.185i q^{67} -148.686i q^{69} -51.2855 q^{71} -346.976 q^{73} -371.192i q^{75} -118.606i q^{77} +173.864 q^{79} +81.0000 q^{81} +524.089i q^{83} +78.0221i q^{85} -534.483 q^{87} +599.736 q^{89} +81.6437i q^{91} +441.686i q^{93} -98.4542 q^{95} +468.989 q^{97} -152.493i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more